Definition:Compound Statement
Definition
A compound statement is a statement which results from the application of one or more logical connectives to a collection of simple statements.
Substatement
A substatement of a compound statement is one of the statements that comprise it.
Ill-Formed Compound Statement
The substatements in a compound statement, which are joined by a connective, may be compound statements themselves.
It is clearly necessary that the interpretation of such a compound statement is unambiguous.
A compound statement is said to be ill-formed if it is ambiguous as to how its substatements are grouped by the action of the connectives.
For example, in natural language:
- I would like some juice or water with ice.
can mean either:
- I would like some juice, or water with ice.
or:
- I would like some juice with ice, or water with ice.
Also defined as
Some sources define a compound statement, in the context of symbolic logic, as a statement form other than:
Such is the approach of 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences, who separately categorises the truth tables of those five statement forms as fundamental.
Hence any statement form which is not one of those fundamental five is (implicitly) defined as being compound.
Also known as
A compound statement can also be referred to as:
Examples
Napoleon
- Napoleon is dead and the world is rejoicing
is a compound statement whose substatements are:
- Napoleon is dead
and:
- The world is rejoicing.
Shape of Eggs
- If all eggs are not square then all eggs are round
is a compound statement whose substatements are:
- All eggs are not square
and:
- All eggs are round.
Barometer
- If the barometer falls then either it will rain or it will snow
is a compound statement whose substatements are:
- The barometer falls
- It will rain
- It will snow.
Also see
- Definition:Statement Form
- Definition:Logical Formula; these represent compound statements in symbolic logic.
- Results about compound statements can be found here.
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.13$: Symbolism of sentential calculus
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.1$: Simple and Compound Statements
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 2$: Logical Constants $(1)$
- 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): compound sentence (molecular sentence)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): compound sentence (molecular sentence)